ArgumentAccording to Hermann von Helmholtz, free mobility of bodies seemed to be an essential condition of geometry. This free mobility can be interpreted either as matter of fact, as a convention, or as a precondition making measurements in geometry possible. Since Henri Poincaré defined conventions as principles guided by experience, the question arises in which sense experiential data can serve as the basis for the constitution of geometry. Helmholtz considered muscular activity to be the basis on which the form of (...) space could be construed. Yet, due to the problem of illusion inherent in the subject’s self-assessment of muscular activity, this solution yielded new difficulties, in that if the manifold is abstracted from rigid bodies which serve as empirical justification of the geometrical notion of space, then illusionary bodies will produce fictive manifolds. The present article is meant to disentangle these difficulties. (shrink)

I analyze the two main theses of Helmholtz's "The Applicability of the Axioms to the Physical World," in which he argued that the axioms of Euclidean geometry are not, as his neo-Kantian opponents had argued, binding on any experience of the external world. This required two argumentative steps: 1) a new account of the structure of our representations which was consistent both with the experience of our (for him) Euclidean world and with experience of a non-Euclidean one, and 2) a (...) demonstration of why geometric propositions are essentially connected to material and temporal aspects of experience. The effect of Helmholtz's discussion is to throw into relief an intermediate category of metrological objects--objects which are required for the properly theoretical activity of doing physical science (in this sense, a priori requirements for doing science), all while being recognizably contingent aspects of experience. (shrink)

The reception of Poincaré’s conventionalist doctrine of space by mathematicians is studied for the period 1891–1911. The opposing view of Riemann and Helmholtz, according to which the geometry of space is an empirical question, is shown to have swayed several geometers. This preference is considered in the context of changing views of the nature of space in theoretical physics, and with respect to structural and social changes within mathematics. Included in the latter evolution is the emergence of non-Euclidean geometry as (...) a new sub-discipline. (shrink)

I said that the book is brilliant. This is not so much because of the conclusions eventually reached about the inadequacy of a purely naturalistic approach to mind. These conclusions are already familiar in the work of Donald Davidson and others. Rather, it is because of the accumulation of historical detail and insight on the basis of which these conclusions are reached. It is often said, for instance, that Kant is a watershed figure, in some sense synthesizing and then moving (...) beyond both empiricism and naturalism. Hatfield uses these terms. But he reinvests them with meaning and energy and in the process shows how radical Kant was in refocusing, by way of a sharp distinction between empirical and transcendental inquiries, the central questions of philosophy. (shrink)